These lectures are mainly based on different joint works with Mehdi Badra.

We consider systems of the form

$$z' = Az + Bf,\quad z(0)=z_0,$$

where A is the infinitesimal generator of an analytic semigroup on a Hilbert space Z, B is a control operator, f ∈ U is the control variable, and U is another Hilbert space. We consider approximate systems

$$z_\varepsilon' = A_\varepsilon z_\varepsilon + B_\varepsilon f,\quad z_\varepsilon(0)=z_{0,\varepsilon},$$

where $A_\varepsilon$ is the infinitesimal generator of an analytic semigroup on a Hilbert space $Z_\varepsilon$, $B_\varepsilon$ is a control operator. For numerical approximations by Finite Element Methods, $\varepsilon =h$ is the mesh size of the triangulation. For approximations by a penalty method, $\varepsilon $ is the penalty parameter.

We are interested in finding feedback gains $K\in{\mathcal L}(Z,U)$ such that $(e^{t(A+BK)})_{t\geq 0}$ is exponentially stable on $Z$. We would like to know if such feedback gains can be approximated by

feedback gains $K_\varepsilon\in{\mathcal L}(Z_\varepsilon,U)$, where $K_\varepsilon$ is chosen so that

$(e^{t(A_\varepsilon+B_\varepsilon K_\varepsilon)})_{t\geq 0}$ is exponentially stable on $Z_\varepsilon$.

We consider the case of nonconforming approximations, that is the case where $Z_\varepsilon \not \subset Z$. We assume that

$Z \subset H$, $Z_\varepsilon \subset H$, $H$ is a Hilbert space, and that there exist projectors $P\in{\mathcal L}(H)$ and

$P_\varepsilon\in{\mathcal L}(H)$ such that $P H = Z$ and $P_\varepsilon H = Z_\varepsilon$.

In Lecture 1, we give sufficient conditions on the triplets $(A,B,P)$ and $(A_\varepsilon,B_\varepsilon,P_\varepsilon)$ and some approximations assumptions with which we prove that Riccati based feedback gains for $(A,B)$ can be approximated by

feedback gains for $(A_\varepsilon,B_\varepsilon)$.

In Lecture 2, we show that the stabilization of the Oseen system (the linearized Navier-Stokes equations around an unstable stationary solution) by a boundary control, and its approximation by a F.E.M. fits into the functional setting introduced in Lecture 1.

We obtain new error estimates for the F.E. approximation of the stationary Oseen system with nonhomogeneous boundary conditions, in convex or non-convex polyhedral domains in ${\mathbb R}^3$.

In Lecture 3, we improve the convergence rates obtained in Lecture 1, by using Reduced Order Model based on spectral projections. We give some applications.

In Lecture 4, we study Fluid-Structure-Interaction systems in the case where the structure is a damped elastic beam or shell located at the boundary of the fluid domain. We recall some existence results of maximal in time strong solutions to these systems. We study the local stabilization of these systems around unstable stationary solutions. We show that the feedback gains can be determined by the method introduced in Lecture 3. Some hints will be given on the numerical approximation of a linearized model.